In the present paper, we consider the cases where the critical exponents \((\ell _1,\ell _2)\)  belong in a subdomain (containing the previous domains) of \([1,\infty )^2\)  ; also, the rotation number is not necessarily Fibonacci type and the results do not depend on the renormalization operator as in [1]}.
We use the formalism presented in [2]} which is based on recursive inequalities analysis of \(\alpha _{n} \) . For technical reason, in the case of bounded geometry, we introduce the vector sequence \(v_n:=(-\ln \alpha _{n}, -\ln \alpha _{n-1})\)  and, the new recursive inequality is controlled by a \(2\times 2\)  matrix. When the rotation number is bi-periodic (\(\rho =[abab\cdots ]; a,b\in {\mathbb {N}}\) ), the \(2\times 2\)  matrix has two eigenvalues (depending on rotation number and critical exponents) \(\lambda _s\in (0,1)\)  and \(\lambda _u>0\) . The equation \(\lambda _u=\lambda _u((a,b); (\ell _1,\ell _2))=1\)  defines a curve \(\mathcal {C}_{\lambda _u=1}\)  (presented above) which
separates the \((\ell _1,\ell _2)\)  plan into two components \({C}_{\lambda _u>1}\)  (below the curve) and \({C}_{\lambda _u<1}\) .
